- algorithm algorithms approximation arithmetic art Babylonian beauty binary binomial coefficients book review bracelets carnival Carnival of Mathematics change chess chess board combinatorics consecutive cookies counting decadic decimal derivatives diagrams divisibility divisors education expansion factorization fibonacci floor foundations fractal fractions game games GIMPS graph graphs Haskell history humor hyperbinary idempotent integers integral interactive intuition irrational irrationality Ivan Niven latex logic Mersenne metric multiplication negative notation number numbers pi prime primes programming proof puzzle P vs NP rectangles repunit review sequence squares triangular video visualization
### Blogroll

### Fun

### Reference

### Categories

- algebra (14)
- arithmetic (35)
- books (25)
- calculus (6)
- challenges (46)
- combinatorics (6)
- complex numbers (3)
- computation (30)
- convergence (9)
- counting (28)
- famous numbers (38)
- fibonacci (14)
- fractals (12)
- games (17)
- geometry (27)
- golden ratio (8)
- group theory (3)
- humor (6)
- induction (7)
- infinity (17)
- iteration (15)
- links (70)
- logic (6)
- meta (37)
- modular arithmetic (10)
- number theory (47)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (54)
- people (19)
- pictures (28)
- posts without words (3)
- primes (23)
- probability (3)
- programming (15)
- proof (42)
- puzzles (10)
- recursion (8)
- review (17)
- sequences (26)
- solutions (24)
- teaching (9)
- trig (3)
- video (18)

### Archives

- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

### Meta

This is somehow related to binary…and yet it’s also related to the Fibonacci Sequence. I’ll have to think about it.

Well, there goes my afternoon. ;)

In keeping with the idea of responding without words:

http://yozh.org/old/images/fib001.png

http://yozh.org/old/images/fib002.png

Neat! Looking at the second image I noticed things I hadn’t thought of when I made mine. =)

It’s almost a cellular automaton with a shift: add a column of green down the right hand side and you get a trace of Wolfram’s Rule # 106/120/169/225 (depending on which color you interpret to be 0 or 1, and on the order in which the cells are traced) with the output shifted one cell to the side at each step.

Very interesting observation! You say it’s very close to a trace of those rules but not exactly. Which makes me wonder whether some sort of cellular automaton can be devised which would give exactly this as its trace — or more likely, would include this as a subtrace, possibly with some extra intermediate states in between.

Edit to the comment I just made: add that column down the right hand side and you get something /very close to/ a trace of one of those rules I mentioned.

I think I have figured out the pattern and I can make pictures like xander has made. But I do not understand the math behind the pattern. Help? Also, why is is tagged “partition”?

If I recall correctly from undergraduate analysis, a partition of a set A is a collection of subsets such that the union of all of the subsets is A, and the intersection of any two subsets is empty. That is, the subsets completely cover the original set, but they do not overlap anywhere.

In the original image, imagine that you are laying stones for a garden path. You have blue stones, which are 1 foot square; and green stones, which are 1 foot by 2 feet. The path is 7 feet long. One question that you might ask is “How many ways can I lay stones to build my path?”

Each possible arrangement of stones represents one partition of the path—the stones completely cover the path, but do not overlap. What I find interesting is that the number of possible partitions of the type pictured is a Fibonacci number (specifically, the

nth Fibonacci number, wherenis the length of the garden path).This can be thought of inductively (or recursively). Let us suppose that there are

F_nways to lay a path that isnfeet long. First, note that there is only one way to pave a 1 foot long path, and that there are two ways to pave a 2 foot long path. ThusF_1 = 1

F_2 = 2.

How many ways are there to lay stones for a path that is

kfeet long? That is, what isF_k? You can either lay a path that isk-1 feet long, and put a 1 foot stone at the end; or you can lay a path that isk-2 feet long, and put a 2 foot stone at the end. This gives us the relationF_k = F_{k-1} + F_{k+2}

which is the Fibonacci sequence.

As an added bonus, it seems that if you add a third type of stone that is 1 foot by 3 feet, you get a sequence defined by

S_1 = 1

S_2 = 1

S_3 = 2

S_n = S_{n-1} + S_{n-2} + S_{n-3}

where S_n is the number of possible partitions for a path of length n.

xander

The solution was good. The explanation was great. Thanks. :)

Instant recognition. I use this problem with students (from middle school through teacher training)

Ghost the Bunny hops up a flight of 7 steps. Ghost can hop up one or two steps at a time, and never hops down. In how many ways can Ghost reach the top step? (3 double hops then a single is not the same as two doubles, a single, and another double)

Yeah, I also instantly thought of the same bijection. I call it the “stair-step”problem.

Pingback: Math Teachers at Play # 39 « Let's Play Math!